Question

solve each equation. Express irrational solutions in exact form and as a decimal rounded to three decimal places :

${\log }_2\left(x^{{\log }_2\left(x\right)}\right)=4$

Short answer

The value of x satisfying the equation is$x=4andx=0.25$

Step-by-step solution

Given information

We are given an equation${\log }_2\left(x^{{\log }_2\left(x\right)}\right)=4$

Simplify

We get,

$$\begin{gathered} {\log }_2\left(x^{{\log }_2\left(x\right)}\right)=4 \\ {\log }_2\left(x\right){\log }_2\left(x\right)=4 \\ {\left({\log }_2\left(x\right)\right)}^2=4{\log }_2\left(x\right)=\pm 2 \end{gathered}$$

Solve for x

We get,

${\log }_2\left(x\right)=\pm 2$

We use the formula $y={\log }_a\left(x\right)\iff x=a^y$

We get

$$\begin{gathered} {\log }_2\left(x\right)=2 \\ x=2^2 \\ x=4 \\ And \\ {\log }_2\left(x\right)=-2 \\ x=2^{-2} \\ x=0.25 \end{gathered}$$